Importing a MySQL database with a new name where the original still exists

So you want to export your MySQL database then import it on the same server with a new name.   I tried this several times and using the SOURCE command inside MySQL or just mysql -p < db.sql both resulted in the database re-importing over the original name.   To fix that run:
sed -i 's/OLDDATABASENAME/NEWDBNAME/g' mysqldumpFile.sql
Then run:  
mysql -p



SOURCE /path/to/sql/file
  No need for a semicolon on the last line.   Then check the success by:  
show tables;